A065474 -id:A065474 - cp's OEIS frontend

A065493 Decimal expansion of the Feller-Tornier constant (1 + A065474)/2.

6, 6, 1, 3, 1, 7, 0, 4, 9, 4, 6, 9, 6, 2, 2, 3, 3, 5, 2, 8, 9, 7, 6, 5, 8, 4, 6, 2, 7, 4, 1, 1, 8, 5, 3, 3, 2, 8, 5, 4, 7, 5, 2, 8, 9, 8, 3, 2, 9, 1, 6, 3, 5, 4, 9, 8, 0, 9, 0, 5, 6, 2, 6, 2, 2, 6, 6, 2, 5, 0, 3, 1, 7, 4, 3, 1, 2, 2, 3, 0, 4, 9, 4, 2, 2, 6, 1, 7, 4, 0, 7, 8, 4, 2, 8, 1, 8, 7

Offset: 0

N. J. A. Sloane, Nov 19 2001

The asymptotic density of numbers with an even number of non-unitary prime divisors (A333634). - Amiram Eldar, May 23 2020

Named after the Croatian-American mathematician William Feller (1906-1970) and the German mathematician Erhard Tornier (1894-1982). - Amiram Eldar, Jun 16 2021

			0.661317049469622335289765846274...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4.1, p. 106.

Jayadev S. Athreya, Cristian Cobeli, and Alexandru Zaharescu, Visibility phenomena in hypercubes, arXiv:2204.03147 [math.NT], 2022.
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107 (1933), pp. 188-232.
Mizan R. Khan and Riaz R. Khan, To count clean triangles we count on imph(n), arXiv:2012.11081 [math.CO], 2020. Mentions this constant.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Feller-Tornier Constant.
Eric Weisstein's World of Mathematics, Prime Products.
Wikipedia, Feller-Tornier constant.

Cf. A065474, A078080, A333634.

digits = 98; r[n_] := -2^n; 1/2 + (1/2) Exp[NSum[r[n]*(PrimeZetaP[2*n]/n), {n, 1, Infinity}, NSumTerms -> 1000, WorkingPrecision -> 2 digits ]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

(1 + prodeulerrat(1 - 2/p^2))/2 \\ Amiram Eldar, Mar 17 2021

A074893 Continued fraction for the Product_{p prime} (1 - 2/p^2) (A065474).

Original entry on oeis.org

0, 3, 10, 19, 2, 1, 2, 2, 1, 6, 1, 6, 19, 17, 1, 7, 1, 2, 2, 1, 10, 2, 6, 2, 1, 3, 2, 1, 21, 5, 1, 15, 1, 1, 4, 1, 1, 1, 443, 2, 1, 4, 3, 1, 1, 6, 26, 6, 2, 39, 4, 1, 2, 6, 1, 1, 2, 4, 7, 1, 5, 1, 3, 1, 3, 5, 10, 1, 9, 5, 1, 2, 4, 10, 1, 1, 5, 1, 1, 3, 2, 2, 2, 2, 1, 4, 1, 1, 1, 1, 11, 1, 4, 1, 2, 2, 2, 54

Offset: 0

Robert G. Wilson v, Sep 13 2002

G. Niklasch, Some number theoretical constants: 1000-digit values

Cf. A007674, A065474.

Increasing partial quotients are in A074178.

(* download the constant from the link above and set it equal to a *) ContinuedFraction[a, 100]

contfrac(prodeulerrat(1- 2/p^2)) \\ Amiram Eldar, Jun 13 2021

a(1) inserted by Amiram Eldar, Jun 13 2021

Offset changed by Andrew Howroyd, Jul 08 2024

A076259 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 4, 2, 2, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 4, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1

Offset: 1

Reinhard Zumkeller, Oct 03 2002

This sequence is unbounded, as a simple consequence of the Chinese remainder theorem. - Thomas Ordowski, Jul 22 2015
Conjecture: lim sup_{n->oo} a(n)/log(A005117(n)) = 1/2. - Thomas Ordowski, Jul 23 2015 [Note: this conjecture is equivalent to lim sup a(n)/log n = 1/2. - Charles R Greathouse IV, Dec 05 2024]
a(n) = 1 infinitely often since the density of the squarefree numbers, 6/Pi^2, is greater than 1/2. In particular, at least 2 - Pi^2/6 = 35.5...% of the terms are 1. - Charles R Greathouse IV, Jul 23 2015
From Amiram Eldar, Mar 09 2021: (Start)
The asymptotic density of the occurrences of 1 in this sequence is density(A007674)/density(A005117) =  A065474/A059956 = 0.530711... (A065469).
The asymptotic density of the occurrences of 2 in this sequence is (density(A069977)-density(A007675))/density(A005117) =  (A065474-A206256)/A059956 = 0.324294...  (End)

			As 24 = 3*2^3 and 25 = 5^2, the next squarefree number greater A005117(16) = 23 is A005117(17) = 26, therefore a(16) = 26-23 = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint, arXiv:2401.13981 [math.NT], 2024.

Cf. A005117, A013661, A020753 (records), A020754, A076260.

Cf. A007674, A007675, A059956, A065469, A065474, A069977, A206256.

a076259 n = a076259_list !! (n-1)
a076259_list = zipWith (-) (tail a005117_list) a005117_list
-- Reinhard Zumkeller, Aug 03 2012

A076259 := proc(n) A005117(n+1)-A005117(n) ; end proc: # R. J. Mathar, Jan 09 2013

Select[Range[200], SquareFreeQ] // Differences (* Jean-François Alcover, Mar 10 2019 *)

t=1; for(n=2,1e3, if(issquarefree(n), print1(n-t", "); t=n)) \\ Charles R Greathouse IV, Jul 23 2015

from math import isqrt
from sympy import mobius
def A076259(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return int(r-m) # Chai Wah Wu, Aug 15 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/6 (A013661). - Amiram Eldar, Oct 21 2020

a(n) < n^(1/5) for large enough n by a result of Pandey. (The constant Pi^2/6 can be absorbed by any eta > 0.) - Charles R Greathouse IV, Dec 04 2024

A007674 Numbers m such that m and m+1 are squarefree.

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 46, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 137, 138, 141, 142, 145

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

m and m+1 squarefree implies that m*(m+1) is a squarefree oblong number and that m*(m+1)/2 is a squarefree triangular number. - Daniel Forgues, Aug 18 2012

Numbers m such that A002378(m) is squarefree. - Thomas Ordowski, Sep 01 2015

P. R. Halmos, Problems for Mathematicians Young and Old. Math. Assoc. America, 1991, p. 28.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, On a problem in additive arithmetic II, Quarterly Journal of Mathematics 3 (1932), pp. 273-290.
Thomas Reuss, Pairs of k-free numbers, consecutive square-full numbers, arXiv:1212.3150 [math.NT], 2012-2014.

Cf. A005117, A013929, A172186, A172187.

ff = {}; gg = {}; Do[kk = FactorInteger[n]; tak = False; Do[If[kk[[m]][[2]] > 1, tak = True], {m, 1, Length[kk]}]; If[tak == False, jj = FactorInteger[n + 1]; tak1 = False; Do[If[jj[[m]][[2]] > 1, tak1 = True], {m, 1, Length[jj]}]; If[tak1 == False, AppendTo[ff, n]]], {n, 1, 500}]; ff (* Artur Jasinski, Jan 28 2010 *)
Select[Range[400],SquareFreeQ[#(#+1)]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)

list(lim)=my(v=vectorsmall(lim\1,i,1),u=List()); for(n=2, sqrt(lim), forstep(i=n^2,lim,n^2, v[i]=v[i-1]=0)); for(i=1,lim, if(v[i], listput(u,i))); v=0; Vec(u) \\ Charles R Greathouse IV, Aug 10 2011

A008966(a(n))*A008966(a(n)+1) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) ~ k*n, where k = 1/A065474. This result is originally due to Carlitz; for the (current) best error term, see Reuss. - Charles R Greathouse IV, Aug 10 2011, expanded Sep 18 2019

Initial 1 added at the suggestion of Zak Seidov, Sep 19 2007

A078147 First differences of sequence of nonsquarefree numbers, A013929.

Original entry on oeis.org

4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, 1, 4, 1, 3, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 1, 3, 1, 1, 2, 4, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 1, 2, 1, 3, 1, 4, 4, 4, 1, 3, 4, 2, 2, 4, 3, 1, 4, 4, 4, 4, 1, 3, 4, 2, 2, 4, 2, 1, 1, 1, 3, 2, 2, 4, 4, 1, 3, 4, 2, 2, 3

Offset: 1

Labos Elemer, Nov 26 2002

Run lengths in A132345, apart from initial run of zeros. - Reinhard Zumkeller, Apr 22 2012

The asymptotic density of the occurrences of 1 in this sequence is density(A068781)/density(A013929) = (1 - 2 * A059956 + A065474)/A229099 = 0.272347... - Amiram Eldar, Mar 09 2021

			a(1) = 4 = 8 - 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A068781, A013929, A132345.

Cf. A059956, A065474, A229099.

a078147 n = a078147_list !! (n-1)
a078147_list = zipWith (-) (tail a013929_list) a013929_list
-- Reinhard Zumkeller, Apr 22 2012

t=Flatten[Position[Table[MoebiusMu[w], {w, 1, 1000}], 0]]; t1=Delete[RotateLeft[t]-t, -1]
Differences[Select[Range[300],!SquareFreeQ[#]&]] (* Harvey P. Dale, May 07 2012 *)

lista(nn) = {my(prec=0); for (n=1, nn, if (!issquarefree(n), if (prec, print1(n-prec, ", ")); prec = n;););} \\ Michel Marcus, Mar 26 2020

from math import isqrt
from sympy import mobius, factorint
def A078147(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values())) # Chai Wah Wu, Sep 10 2024

a(n) = A013929(n+1) - A013929(n).
a(n) = 1, 2, 3 or 4 since n = 4*k is always nonsquarefree.
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/(Pi^2-6) = 2.550546... - Amiram Eldar, Oct 21 2020

Offset fixed by Reinhard Zumkeller, Apr 22 2012

A072284 Numbers k begins a new chain of squarefree integers. I.e., k is squarefree but k-1 is not.

Original entry on oeis.org

1, 5, 10, 13, 17, 19, 21, 26, 29, 33, 37, 41, 46, 51, 53, 55, 57, 61, 65, 69, 73, 77, 82, 85, 89, 91, 93, 97, 101, 105, 109, 113, 118, 122, 127, 129, 133, 137, 141, 145, 149, 151, 154, 157, 161, 163, 165, 170, 173, 177, 181, 185, 190, 193, 197, 199, 201, 205, 209

Offset: 1

Joseph L. Pe, Jul 10 2002

The asymptotic density of this sequence is 1/zeta(2) - Product_{p prime} (1 - 2/p^2) = A059956 - A065474 = 0.2852930029... (Matomäki et al., 2016) - Amiram Eldar, Feb 14 2021

			1 begins a new chain 1, 2, 3 of squarefree integers. 4 is not squarefree. Then 5 begins a new chain 5, 6, 7 of squarefree integers. Hence 1 and 5 are terms of the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A001221, A005117, A013929, A120992, A136742, A136743.

Select[Range[100], MoebiusMu[# - 1] == 0  && Abs[MoebiusMu[#]] == 1 &] (* Amiram Eldar, Feb 14 2021 *)
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,0,250}],{0,1}][[All,2]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 24 2021 *)

n=1; for(k=1,100, while(!issquarefree(n),n=n+1); print1(n","); while(issquarefree(n),n=n+1))

From Reinhard Zumkeller, Jan 20 2008: (Start)
A136742 mod a(n) = 0;
A136742(n) = Product_{k=0..A120992(n)-1} (a(n) + k);
A136743(n) = Sum_{k=0..A120992(n)-1} A001221(a(n) + k). (End)

More terms from Ralf Stephan, Mar 19 2003

A068781 Lesser of two consecutive numbers each divisible by a square.

Original entry on oeis.org

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024

			44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From _Gus Wiseman_, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
  (4,8)
  (9,12,16,18,20,24)
  (25,27)
  (28,32,36,40,44)
  (45,48)
  (49)
  (50,52,54,56,60,63)
  (64,68,72,75)
  (76,80)
  (81,84,88,90,92,96,98)
  (99)
The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
Cf. A049535, A077647, A078143, A045882.
Subsequence of A261869.
Cf. A007674, A373409, A373410, A373412, A375707, A376164.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.

a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015

Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)

isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021

A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015

A073247 Squarefree numbers k such that k-1 and k+1 are not squarefree.

Original entry on oeis.org

17, 19, 26, 51, 53, 55, 89, 91, 97, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 349, 362, 377, 379, 413, 415, 449, 451, 485, 487, 489, 491, 521, 523, 530, 551, 557, 559, 577, 579, 593, 595

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Probably 11*n < a(n) < 12*n for n > 189. - Charles R Greathouse IV, Nov 05 2017

The asymptotic density of this sequence is 1/zeta(2) - 2 * Product_{p prime} (1 - 2/p^2) + Product_{p prime} (1 - 3/p^2) = A059956 - 2*A065474 + A206256 = 0.088145884881346585838... . - Amiram Eldar, Aug 30 2024

Christopher E. Thompson, Table of n, a(n) for n = 1..10000

Cf. A005117, A073249, A073248, A007675.
Cf. A268331, A268332, A268333, A268334 (squarefree numbers isolated by more than 2, 3, etc.).
Cf. A059956, A065474, A206256.

sf:= select(numtheory:-issqrfree,[$1..1000]):
map(t -> `if`(sf[t-1]=sf[t]-1 or sf[t+1]=sf[t]+1,NULL,sf[t]), [$2..nops(sf)-1]); # Robert Israel, Feb 01 2016

Reap[For[n = 0, n <= 1000, n++, If[SquareFreeQ[n] && !SquareFreeQ[n-1] && !SquareFreeQ[n+1], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 26 2019 *)

is(n)=!issquarefree(n-1) && issquarefree(n) && !issquarefree(n+1) \\ Charles R Greathouse IV, Nov 05 2017

list(lim)=my(v=List(),l1,l2); forfactored(k=9,lim\1+1, if(!issquarefree(k) && !issquarefree(l2) && issquarefree(l1), listput(v,l1[1])); l2=l1; l1=k); Vec(v) \\ Charles R Greathouse IV, Nov 27 2024

A058026 Number of positive integers, k, where k <= n and gcd(k,n) = gcd(k+1,n) = 1.

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 5, 0, 3, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 15, 0, 9, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 9, 0, 45, 0, 35, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 15, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 15, 0, 45, 0, 77, 0, 27, 0, 81, 0, 45, 0, 27, 0

Offset: 1

Leroy Quet, Nov 15 2000

Called the Schemmel totient function in the Handbook of Number Theory II. - R. J. Mathar, Apr 15 2011
a(n) is also the number of units u in Z/nZ such that Phi(1,u) or Phi(2,u) is a unit, where Phi is the cyclotomic polynomial. - Jordan Lenchitz, Jul 12 2017
This is the function phi(n, 1) in Alder. - Michel Marcus, Nov 14 2017

			a(15) = 3 because 1 and 2, 7 and 8 and 13 and 14 are all relatively prime to 15.
a(15) = a(3*5) = a(3)*a(5) = 3^0*(3-2)*5^0*(5-2) = 3.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 276.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Enrique Pérez Herrero)
Henry L. Alder, A Generalization of the Euler phi-Function, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 690-692.
O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Colin Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015) # 15.2.1.
Leonard Eugene Dickson, Schemmel's Generalization of Euler's phi-Function, History of the Theory of Numbers, Vol. 1: Divisibility and Primality, Washington, Carnegie Institution of Washington, 1919, p. 147.
Mizan R. Khan and Riaz R. Khan, To count clean triangles we count on imph(n), arXiv:2012.11081 [math.CO], 2020.
Walter Klotz and Torsten Sander, Some Properties of Unitary Cayley Graphs, The Electronic Journal of Combinatorics, Volume 14 (2007), #R45. See Corollary 7 p. 4.
Emma T. Lehmer, A numerical function applied to cyclotomy, Bull. Amer. Math. Soc. 36 (1930), 291-298.
Victor Schemmel, Ueber relative Primzahlen, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 191-192.

Cf. A070554, A069828, A027748, A124010, A289460.

Cf. A000010 (phi(n,0)), A002472 (phi(n,2)).

a058026 n = product $ zipWith (\p e -> p ^ (e - 1) * (p - 2))
                              (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, May 10 2014

A058026 := proc(n) local a; a := n ; for p in numtheory[factorset](n) do a := a*(1-2/p) ; end do: a ; end proc: # R. J. Mathar, Apr 15 2011

a[n_] := DivisorSum[n, n/# MoebiusMu[#] DivisorSigma[0, #]&]; Array[a, 90] (* Jean-François Alcover, Dec 05 2015 *)
f[p_, e_] := (p-2) * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

a(n) = sumdiv(n, d, n/d*moebius(d)*numdiv(d)); \\ Michel Marcus, Apr 27 2014

a(n) = n*prod(p=1, n, if (isprime(p) && !(n % p), (1-2/p), 1)); \\ Michel Marcus, Feb 02 2016

a(n) = my(r=1,f=factor(n)); for(j=1, #f[,1], my(p=f[j,1],e=f[j,2]); r*=(p-2)*p^(e-1)); return(r); \\ Jianing Song, Nov 01 2022

Multiplicative with a(p^e) = p^(e-1)*(p-2). - Vladeta Jovovic, Dec 01 2001
a(n) = Sum_{d|n} n/d*mu(d)*tau(d). - Vladeta Jovovic, Apr 29 2002
a(n) = Sum_{d divides n} phi(n/d)*(-1)^omega(d). - Vladeta Jovovic, Sep 26 2002
A003557(n) | a(n). - R. J. Mathar, Mar 30 2011
a(n) = n*Product_{primes p|n} (1-2/p). Dirichlet g.f. zeta(s-1)*product_p (1-2*p^(-s)). - R. J. Mathar, Apr 15 2011
a(n) = phi(n) * Sum_{d|n} mu(d)/phi(d), where mu(k) is the Moebius function and phi(k) is the Euler totient function. - Daniel Suteu, Jun 23 2018
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065474 = Product_{primes p} (1 - 2/p^2) = 0.32263409893924467057953169254823706657095... - Vaclav Kotesovec, Dec 18 2019, corrected May 22 2025 (typo found by Amiram Eldar)
a(n) = Sum_{k=1..n} (-1)^omega(gcd(n,k)). - Ilya Gutkovskiy, Feb 22 2020
a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2)*floor(n/(d1*d2)). - Ridouane Oudra, Dec 31 2022

A335963 Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).

Original entry on oeis.org

8, 9, 4, 8, 4, 1, 2, 2, 4, 5, 6, 2, 4, 8, 8, 1, 7, 0, 7, 2, 5, 6, 6, 1, 5, 0, 6, 9, 0, 8, 4, 3, 7, 3, 2, 1, 9, 8, 7, 5, 4, 7, 8, 0, 8, 9, 2, 0, 7, 1, 8, 9, 7, 2, 6, 0, 1, 7, 9, 9, 4, 2, 7, 6, 1, 6, 5, 6, 3, 8, 9, 2, 2, 1, 2, 0, 9, 1, 5, 5, 0, 2, 8, 8, 5, 9, 4, 2, 9, 1, 0, 5, 3, 9, 5, 8, 9, 1, 0, 8, 0, 0, 3, 3, 2, 2

Offset: 0

Amiram Eldar, Jul 01 2020

The asymptotic density of the numbers k such that k^2+1 is squarefree (A049533) (Estermann, 1931).
The constant c in Sum_{k=0..n} phi(k^2 + 1) = A333170(n) ~ (1/4)*c*n^3 (Finch, 2018).
The constant c in Sum_{k=0..n} phi(k^2 + 1)/(k^2 + 1) = (3/4)*c*n + O(log(n)^2) (Postnikov, 1988).

			0.89484122456248817072566150690843732198754780892071...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 101.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 166.
A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., 1988, pp. 192-195.

Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Mathematische Annalen, Vol. 105 (1931), pp. 653-662, alternative link.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 14.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica, Vol. 155, No. 1 (2012), pp. 1-13.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], variable F(m=4,n=1,s=2), p. 38.
Wolfgang Schwarz, Über die Summe Sigma_{n <= x} phi(f)(n) und verwandte Probleme, Monatshefte für Mathematik, Vol. 66, No. 1 (1962), pp. 43-54, alternative link.
Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222.

Cf. A002144, A049533, A065474, A069987, A088539, A333169, A333170, A340617.

Digits := 150;
with(NumberTheory);
DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
pzetamod41 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(alfa((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
evalf(exp(-Sum(2^t*pzetamod41(2*t, 50)/t, t = 1..200))); # Vaclav Kotesovec, Jan 13 2021

f[p_] := If[Mod[p, 4] == 1, 1 - 2/p^2, 1]; RealDigits[N[Product[f[Prime[i]], {i, 1, 10^6}], 10], 10, 8][[1]] (* for calculating only the first few terms *)
(* -------------------------------------------------------------------------- *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z2[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = 2^w * P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

f(lim,poly=1-'x-'x^2/2)=prodeulerrat(subst(poly,'x,1/'x^2))*prodeuler(p=2,lim, my(pm2=1./p^2); if(p%4==1,1.-2*pm2,1.)/subst(poly,'x,pm2)) \\ Gets 14 digits at lim=1e9; Charles R Greathouse IV, Aug 10 2022

Equals 2*A065474/A340617.

More digits (from the paper by R. J. Mathar) added by Jon E. Schoenfield, Jan 12 2021

More digits from Vaclav Kotesovec, Jan 13 2021

cp's OEIS Frontend

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